In mathematics, Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials, introduced by Pafnuty Chebyshev and rediscovered by Wolfgang Hahn. The Hahn class is a name for special cases of Hahn polynomials, such as Meixner polynomials, Krawtchouk polynomials, and Charlier polynomials. Sometimes the Hahn class is taken to include limiting cases of these polynomials, in which case it also includes the classical orthogonal polynomials.
Hahn polynomials are defined in terms of generalized hypergeometric functions by
for . If , these polynomials are identical to the discrete Chebyshev polynomials up to a scale factor.
Hahn polynomials have dual, continuous and continuous dual versions. These polynomials all have q-analogs. Hahn polynomials are a limiting case of Racah polynomials.
Properties Orthogonality Hahn polynomials satisfy the orthogonality relations
where is the Kronecker delta and the weight functions are
and
Dual version Hahn polynomials have a dual version, the dual Hahn polynomials, which are also a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined on a non-uniform lattice by
for , and the parameters are restricted to
Here is the rising factorial, and is a generalized hypergeometric function.
give a detailed list of their properties.
Orthogonality The dual Hahn polynomials satisfy the orthogonality condition
for , where ,
and
Numerical instability As increases, the values that the discrete polynomials attain increase. As a result, obtaining numerical stability in calculating the polynomials requires the use of renormalized dual Hahn polynomials, defined as
for . Here the orthogonality condition becomes
for
Continuous version Hahn polynomials have a continuous version, the continuous Hahn polynomials, which are also a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by
give a detailed list of their properties.
Orthogonality The continuous Hahn polynomials are orthogonal with respect to the weight function
In particular, they satisfy the orthogonality relation
for , , , , , .
Recurrence and difference relations The sequence of continuous Hahn polynomials satisfies the recurrence relation
where
and
Rodrigues formula The continuous Hahn polynomials are given by the Rodrigues-type formula
Generating functions The continuous Hahn polynomials have the following generating function:
A second, distinct generating function is given by
Relation of continuous Hahn polynomials to other polynomials
The Wilson polynomials are a generalization of the continuous Hahn polynomials. The Bateman polynomials Fn(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by
The Jacobi polynomials Pn(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:Koekoek, Lesky, & Swarttouw (2010), p. 203.
Continuous dual Hahn polynomials In mathematics, the continuous dual Hahn polynomials are a family of orthogonal polynomials in the Askey scheme of hypergeometric orthogonal polynomials. They are defined in terms of generalized hypergeometric functions by
give a detailed list of their properties.
Closely related polynomials include the dual Hahn polynomials Rn(x;γ,δ,N), the continuous Hahn polynomials pn(x,a,b, , ), and the Hahn polynomials. These polynomials all have q-analogs with an extra parameter q, such as the q-Hahn polynomials Qn(x;α,β, N;q), and so on.
Relation of continuous dual Hahn polynomials to other polynomials
Wilson polynomials are a generalization of continuous dual Hahn polynomials
q-Hahn polynomials In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions by
Dual q-Hahn polynomials In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition The polynomials are given in terms of basic hypergeometric functions.
Continuous q-Hahn polynomials In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by Roelof p433, Springer 2010
Gallery
Relation of q-Hahn polynomials to other polynomials
q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：
q-Hahn polynomials→ Hahn polynomials
make the substitution, into definition of q-Hahn polynomials, and find the limit q→1, we obtain
，which is exactly Hahn polynomials.
Continuous dual q-Hahn polynomials In mathematics, the continuous dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by Mesuma Atakishiyeva, Natig Atakishieyev, A NON STANDARD GENERATING FUNCTION FOR CONTINUOUS DUAL Q-HAHN POLYNOMIALS, REVISTA DE MATEMATICA 2011 18(1):111-120
In which
Gallery
Relation of Hahn polynomials to other polynomials
Racah polynomials are a generalization of Hahn polynomials
Notes
References